WBBSEClass XMathematicsPythagoras Theorem05 Mark2025 Moderate

QQuestion

State and prove Pythagoras theorem.

Exam-ready answer • 05 Mark

Answer

Proof of pythagoras Theorem 2 removebg preview e1733890938299

Pythagoras Theorem: In any right-angled triangle, the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the other two sides.

Given: ABC is a right-angled triangle whose ∠A is a right angle.

To prove: BC² = AB² + AC²

Construction: I draw a perpendicular AD on the hypotenuse BC from the right angular point A which intersects the side BC at the point D.

Proof: In right-angled Δ ABC, AD is perpendicular on the hypotenuse BC.

∴ Δ ABD and Δ CBA are similar.

Hence,

{\text{AB} \over \text{BC}} = {\text{BD} \over \text{AB}},

∴ AB² = BC × BD —- (i)

Again, ΔCAD and ΔCBA are similar.

Hence,

{\text{AC} \over \text{BC}} = {\text{DC} \over \text{AC}}

∴ AC² = BC × DC —- (ii)

So, by adding (i) and (ii), I get:

AB² + AC² = BC × BD + BC × DC.

= BC (BD + DC) = BC × BC = BC².

∴ BC² = AB² + AC² [proved]

Was this answer helpful?

Related Questions

More WBBSE Moderate level questions

1/12

O is a point inside a rectangle ABCD such that OB = 6 cm, OD = 8 cm and OA = 5 cm. Find OC.

Moderate 02 Mark

In △ABC, ∠ABC = 90°, AB = 6 cm, BC = 8 cm. Find the circum-radius of triangle ABC.

Moderate 02 Mark

In a triangle PQR, ∠P = 90° and PS is perpendicular to QR. Then prove that

Hard 03 Mark

Distance between two pillars is 150 m, height of one pillar is three times the other pillar. From the mid-point of the line joining their foot the angles of elevation of the top of the two pillars are complementary. Find the height of the smaller pillar

Moderate 05 Mark

From the middle of a ground Habu observes a flying bird toward north at an angle of elevation 30°, after 2 minutes he observes the same bird at an angle of elevation 60° toward south. If the bird always flies in a straight line at height 50√3 meters from the ground level, find its velocity.

Moderate 05 Mark

Draw a triangle whose sides are of lengths 6 cm, 8 cm, and 10 cm. Draw the incircle of this triangle.

Moderate 05 Mark

Find the value of 2√3 geometrically.

Moderate 05 Mark

AB is the diameter of the circle with centre O. From a point P on the circle, a perpendicular PN is drawn on AB. Prove geometrically that PB² = AB.BN.

Moderate 05 Mark

Prove that in a cyclic quadrilateral opposite angles are supplementary.

Moderate 05 Mark

Divide ₹ 21,866 into two parts such that the amount of the first part for 3 years and amount of the second part for 5 years are equal at the same rate of 5% compound interest.

Moderate 05 Mark

In a joint business the Capital of Samar and Mohim are ₹ 20,000 each. After 6 months Samar invests ₹ 5,000 more, but Mohim withdraws ₹ 5,000. At the end of the year if the profit is ₹ 32,000 then find the share of the profit of Samar and Mohim.

Moderate 05 Mark

Find the ratio of length of radius and perimeter of a circle

Moderate 01 Mark